Reduced Power Graphs of ${\mathrm{PGL}}_3({\mathbb{F}}_q)$

Abstract

Given a group $G$, let us connect two non-identity elements by an edge if and only if one is a power of another. This gives a graph structure on $G$ minus identity, called the reduced power graph. In this paper, we shall find the exact number of connected components and the exact diameter of each component for the reduced power graphs of ${\mathrm{PGL}}_3({\mathbb{F}}_q)$ for all prime power $q$.